NON-WELL-FOUNDED SET BASED MULTI-AGENT EPISTEMIC ACTION LANGUAGE 34 - PowerPoint PPT Presentation

University of Udine Department of Mathematics, Computer Science and Physics NON-WELL-FOUNDED SET BASED MULTI-AGENT EPISTEMIC ACTION LANGUAGE 34 th Italian Conference on Computational Logic Francesco Fabiano , Idriss Riouak, Agostino Dovier and Enrico Pontelli June 20, 2019

Overview 1 1. Multi-Agent Epistemic Planning 2. Kripke Structures 3. Possibilities 4. The action language m A ρ 5. Conclusions Francesco Fabiano , Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

Chapter 1 Multi-Agent Epistemic Planning Francesco Fabiano , Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

Multi-Agent Epistemic Planning Introduction 2 Epistemic Reasoning Reasoning not only about agents’ perception of the world but also about agents’ knowledge and/or beliefs of her and others’ beliefs. Francesco Fabiano , Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

Multi-Agent Epistemic Planning Introduction 2 Epistemic Reasoning Reasoning not only about agents’ perception of the world but also about agents’ knowledge and/or beliefs of her and others’ beliefs. Multi-agent Epistemic Planning Problem [BA11] Finding plans where the goals can refer to: - the state of the world - the knowledge and/or the beliefs of the agents Francesco Fabiano , Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

Multi-Agent Epistemic Planning An Example 3 Francesco Fabiano , Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

Multi-Agent Epistemic Planning An Example 3 Initial State - Snoopy and Charlie are looking while Lucy is ¬ looking - No one knows the coin position . Francesco Fabiano , Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

Multi-Agent Epistemic Planning An Example 4 Goal State - Charlie knows the coin position - Lucy knows that Charlie knows the coin position - Snoopy does not know anything about the plan execution Francesco Fabiano , Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

Multi-Agent Epistemic Planning Challenges 5 An agent has to reason about his actions effects on - The state of the world - The agents ’ awareness of the environment - The agents ’ awareness of other agents ’ actions - The knowledge of other agents about his own Francesco Fabiano , Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

Multi-Agent Epistemic Planning Notations 6 Given a set of agents AG Modal operator B ag where ag ∈ AG Models the beliefs of ag about the state of the world and/or about the beliefs of other agents . Francesco Fabiano , Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

Multi-Agent Epistemic Planning Notations 6 Given a set of agents AG Modal operator B ag where ag ∈ AG Models the beliefs of ag about the state of the world and/or about the beliefs of other agents . Group operator C α where α ⊆ AG Expresses the common belief of a group of agents . Francesco Fabiano , Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

Multi-Agent Epistemic Planning Notations 6 Given a set of agents AG Modal operator B ag where ag ∈ AG Models the beliefs of ag about the state of the world and/or about the beliefs of other agents . Group operator C α where α ⊆ AG Expresses the common belief of a group of agents . Belief Formulae Take into consideration fluents and/or agents ’ beliefs. Francesco Fabiano , Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

Multi-Agent Epistemic Planning Example of Belief Formulae 7 Given - AG = { Snoopy , Charlie , Lucy } - F = { opened , head , looking ag } ag ∈ AG B Snoopy B Charlie ¬ opened Snoopy believes that Charlie believes that the box is ¬ opened . Francesco Fabiano , Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

Multi-Agent Epistemic Planning Example of Belief Formulae 7 Given - AG = { Snoopy , Charlie , Lucy } - F = { opened , head , looking ag } ag ∈ AG B Snoopy B Charlie ¬ opened Snoopy believes that Charlie believes that the box is ¬ opened . C α ( ¬ B Lucy heads ∧ ¬ B Lucy ¬ head ) where α = AG It is common knowledge that Lucy does not know whether the coin lies heads or tails up Francesco Fabiano , Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

Multi-Agent Epistemic Planning Knowledge vs. Belief 8 - The modal operator B ag represents the worlds’ relation - Different relation’s properties imply different meaning for B ag Francesco Fabiano , Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

Multi-Agent Epistemic Planning Knowledge vs. Belief 8 - The modal operator B ag represents the worlds’ relation - Different relation’s properties imply different meaning for B ag - K nowledge and B elief are characterized by a subset of the following axioms Serial ( D ) and S5 ( K,T,4,5 ) Axioms Given the fluent formulae φ , ψ and the worlds i , j D ¬R i ⊥ B K K ( R i ϕ ∧ R i ( ϕ ⇒ ψ )) ⇒ R i ψ B K T R i ϕ ⇒ ϕ K 4 R i ϕ ⇒ R i R i ϕ B K 5 ¬R i ϕ ⇒ R i ¬R i ϕ B K Francesco Fabiano , Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

Chapter 2 Kripke Structures Francesco Fabiano , Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

Kripke Structures Description 9 Pointed Kripke structure A Pointed Kripke structure is a pair ( � S , π, R 1 , . . . , R n � , s 0 ), s.t.: - S is a set of worlds and s 0 ∈ S - π : S �→ 2 F associates an interpretation to each element of S - for 1 ≤ i ≤ n, R i ⊆ S × S is a binary relation over S Francesco Fabiano , Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

Kripke Structures Description 9 Pointed Kripke structure A Pointed Kripke structure is a pair ( � S , π, R 1 , . . . , R n � , s 0 ), s.t.: - S is a set of worlds and s 0 ∈ S - π : S �→ 2 F associates an interpretation to each element of S - for 1 ≤ i ≤ n, R i ⊆ S × S is a binary relation over S ag ag ag = { Charlie , Lucy , Snoopy } ag i 1 i 0 { Snoopy } { Snoopy } { Snoopy } { Snoopy } { Charlie , Lucy } { Charlie , Lucy } { Lucy } g 0 g 1 Francesco Fabiano , Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

Kripke Structures Entailment 10 Let ϕ be a belief formula and ( M , s) be a pointed Kripke structure: Entailment w.r.t. a pointed Kripke structure - ( M , s) | = ϕ if ϕ is a fluent formula and π (s) | = ϕ ; Francesco Fabiano , Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

Kripke Structures Entailment 10 Let ϕ be a belief formula and ( M , s) be a pointed Kripke structure: Entailment w.r.t. a pointed Kripke structure - ( M , s) | = ϕ if ϕ is a fluent formula and π (s) | = ϕ ; - ( M , s) | = B ag i ϕ if ∀ t: (s , t) ∈ R i it holds that ( M , t) | = ϕ ; Francesco Fabiano , Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

Kripke Structures Entailment 10 Let ϕ be a belief formula and ( M , s) be a pointed Kripke structure: Entailment w.r.t. a pointed Kripke structure - ( M , s) | = ϕ if ϕ is a fluent formula and π (s) | = ϕ ; - ( M , s) | = B ag i ϕ if ∀ t: (s , t) ∈ R i it holds that ( M , t) | = ϕ ; - ( M , s) | = E α ϕ if ( M , s) | = B ag i ϕ for all ag i ∈ α ; = E k - ( M , s) | = C α ϕ if ( M , s) | α ϕ for every k ≥ 0, where E 0 α ϕ = ϕ and E k +1 ϕ = E α ( E k α ϕ ). α The entailment for the standard operators is defined as usual Francesco Fabiano , Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

Kripke Structures Problems 11 - Solvers require high amount of memory - In literature the states have been represented explicitly - State comparison needs to find bisimilar states { B , C } { B , C } r 1 r 1 { B , C } { A } { A } q 0 { A } { A } { A } { A } { A , B , C } { A , B , C } { A , B , C } { A , B , C } s 0 { A , B , C } s 1 s 0 { A , B , C } s 1 Francesco Fabiano , Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

Kripke Structures Solutions 12 - Heuristics [Le+18] Francesco Fabiano , Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

Kripke Structures Solutions 12 - Heuristics [Le+18] - Symbolic representation of Kripke structures Francesco Fabiano , Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

Kripke Structures Solutions 12 - Heuristics [Le+18] - Symbolic representation of Kripke structures - Alternative representations Francesco Fabiano , Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

Chapter 3 Possibilities Francesco Fabiano , Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

Possibilities Overview 13 - Introduced by Gerbrandy and Groeneveld [GG97] - Used to represent multi-agent information change - Based on non-well-founded sets - Corresponds with a class of bisimilar Kripke structures [Ger99] Francesco Fabiano , Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

Possibilities Non-well-founded sets theory Non-well-founded sets 14 Non-well-founded set [Acz88] A set is non-well-founded (or extraordinary ) when among its descents there are some which are infinite. The non-well-founded set Ω = { Ω } Francesco Fabiano , Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC